The Economics of Space 433: Section 5 Normalization and Log-Linearization · 2020. 2. 28. ·...

The Economics of Space 433: Section 5

Normalization and Log-Linearization

Costas Arkolakis and Eduardo Fraga1

1Yale University

27 February 2020

The Economics of Space. Lecture 8-9: c© Costas Arkolakis and Eduardo Fraga 1

Goals

I Our goal in these slides is twofold:I First, we will discuss why our workhorse spatial model features one price

normalizationI Second, we will work on the algebra of log-linearization.

I Price normalization:I Equilibrium system has one less equation than it looks (courtesy of Walras’s Law)I We then impose one price normalization to ‘‘close’’ the system.

I Log-linearization:I Express shocks to the equilibrium system and the response of endogenous

variables...I ... but in a compact way.

The Economics of Space. Lecture 8-9: Introduction c© Costas Arkolakis and Eduardo Fraga 2

Roadmap

I Normalization

I Log-Linearization


The Equilibrium System

I Consider our workhorse spatial model with N locations

I Given exogenous variables {τij , Āi , ūi} and parameters (α, β, σ), the equilibriumendogenous variables {Li ,wi ,Pi , W̄ } are found by solving the following system ofequations:

wiLi =∑j

(τijwi

ĀiLαi Pj

)1−σwjLj (1)

W̄ =wiPi

ūiL−βi (2)∑

i

Li = L̄ (3)

Pi =

(∑k

(τkiwk

ĀkLαk

)1−σ) 11−σ(4)


Equations and Unknowns

I Counting the number of equations in the system:I Equation (1) features N equations (one for each location i = 1, 2, ...,N)I Same for equations (2) and (4)I Equation (3) features only one equation.I Total: 3N + 1

I Counting the number of unknowns (i.e. endogenous variables) in the system:I Variable wi features N unknowns, one for each location: w1,w2,, ...,wNI Same for variables P and LI Variable W̄ features only one unknown (since welfare is the same everywhere)I Total: 3N + 1

I Thus we have an equal number of equations and unknowns:I Suggests that the system has an unique solution for the set of endogenous

variables {wi , Li ,Pi , W̄ }


Walras’s Law

I But wait! We actually have one fewer equation than it appears.I The feasibility equation (1) actually features N − 1 equations, not N!

I Why? Walras’s Law

I A principle formulated by Leon Walras in 1874

I Important Corollary of the Law:I In an economy, if all markets but one are in equilibrium, then that last market

must also be in equilibriumI For proof of Corollary, check Appendix slides


Walras’s Law in the Spatial Model

I Apply Corollary to our spatial model’s feasibility condition (equation (1)):I If N − 1 markets are in equilibrium, then the Nth market must also be in

equilibriumI Thus, if equation (1) holds for i = 1, 2, ...,N − 1, then it must also hold for i = NI Nth equation doesn’t add any new information!

I We thought we had N feasibility equations, but we actually have only N − 1I The Nth equation is redundant!

I Thus, our equilibrium system only really has 3N equations, not 3N + 1I Which means we have fewer equations (3N) than unknowns (3N + 1)I System of equations has a potentially infinite number of solutions!


Closing the Model: Price Normalization

I Therefore, we need one extra equation to ‘‘pin down’’ a single solution from thepotentially infinite set of solutions.

I That’s where price normalization comes in

I Pick a monetary variable and impose an arbitrarily chosen value to it:I For example, impose w1 = 1 (a practical choice)I Note that other alternative choices would also be valid, e.g. w32 = 12.93

√37 (less

practical choice but still valid)

I With this extra price-normalization equation, we are back to 3N + 1 equationsI The equation system should now have a single solution.I We may now be able to find that solution, either analytically or using numerical

methods.


Roadmap

I Normalization

I Log-Linearization


Log-Linearization

I In Lecture 13-14 we derived the following equation:

X̃i = W1−σ

∑j

(τij/Āi )1−σX̃j (5)

I Applying logs and totally differentiating, we will arrive at the following equation:

d ln X̃i −∑j

yijd ln X̃j − (1− σ)d lnW = (1− σ)∑j

yijd ln(τij/Āi ) (6)

I This process is called log-linearizationI Equation system becomes expressed in terms of ‘‘changes’’ (hence the d operators)

rather than levels

I Next, we show how to go from equation (5) to equation (6), step by step.


A Few Algebraic Facts

I Before we start, let’s remember a few algebraic facts about logarithms:

(F1) : ln(ab) = ln(a) + ln(b)

(F2) :d ln(a)

da=

1

a

(F3) : da = a× d ln(a)

(F4) : d ln(a) =da

a

(F5) : d(m + n) = dm + dn

(F6) : ln ab = b × ln(a)

I Comments:I Note that (F3) and (F4) are consequences of (F2)I (F5) is reminiscent of the fact that the derivative is a linear operator:

d(f (x)+g(x))dx

= df (x)dx

+ dg(x)dx


Log-Linearization (1)

I Let’s start the log-linearization process. First take logs of equation (5) and usefacts (F1) and (F6):

ln X̃i = (1− σ) lnW + ln(∑

j

(τij/Āi )1−σX̃j

)I Now take differentials of both sides and use (F5):

d ln X̃i = (1− σ)d lnW + d ln(∑

j

(τij/Āi )1−σX̃j

)I Apply (F4) to the latter term:

d ln X̃i = (1− σ)d lnW +d(∑

j(τij/Āi )1−σX̃j

)∑

j′(τij′/Āi )1−σX̃j′

I Apply (F5) to bring the differential operator into the parentheses:

d ln X̃i = (1− σ)d lnW +

∑j d[(τij/Āi )

1−σX̃j]

∑j′(τij′/Āi )

1−σX̃j′The Economics of Space. Lecture 8-9: Introduction c© Costas Arkolakis and Eduardo Fraga 12


I Then use (F3) to transform the term inside the square brackets:


∑j

[(τij/Āi )

1−σX̃j

]× d ln

[(τij/Āi )

1−σX̃j

]∑

j′(τij′/Āi )1−σX̃j′

I Now apply (F1), (F6), and (F5) in succession to term d ln

[(τij/Āi )

1−σX̃j

]to

obtain:


∑j

[(τij/Āi )

1−σX̃j

]×[

(1− σ)d ln(τij/Āi ) + d ln X̃j]

∑j′(τij′/Āi )

1−σX̃j′



I Note that the denominator of the ‘‘big fraction’’, namely∑

j′(τij′/Āi )1−σX̃j′ ,

doesn’t depend on j .I Let us then bring it inside the ‘‘

∑j ’’ term:

d ln X̃i = (1−σ)d lnW+∑j

((τij/Āi )

1−σX̃j∑j′(τij′/Āi )

1−σX̃j′

)×[

(1−σ)d ln(τij/Āi )+d ln X̃j]

I Let’s now define term yij ≡ (τij/Āi )1−σX̃j∑

j′ (τij′/Āi )1−σX̃j′

. This greatly simplifies notation:

d ln X̃i = (1− σ)d lnW +∑j

yij [(1− σ)d ln(τij/Āi ) + d ln X̃j ]



I Distributing yij across the terms inside the square brackets and breaking up thesummation, we get:

d ln X̃i = (1− σ)d lnW +∑j

yij(1− σ)d ln(τij/Āi ) +∑j

yijd ln X̃j

.

I Now we just have to rearrange terms across the two sides of the equation to get:

d ln X̃i −∑j

yijd ln X̃j − (1− σ)d lnW = (1− σ)∑j

yijd ln(τij/Āi )

I This is the same as equation (6)I We’ve achieved our log-linearization goal!


Roadmap

I Normalization

I Log-Linearization

I Appendix


Appendix

I In this Appendix, we prove the Corollary of Walras’s Law

I Namely, we prove that if all markets but one are in equilibrium, then that lastmarket must also be in equilibrium

I Formal statement:I If equation (1) holds for i = 1, 2, ...,N − 1, then it must also hold for i = N


Proving the Corollary (1)

I For conciseness of notation, define the variable λij ≡(

τijwiĀiLαi Pj

)1−σI For a given destination j , summing across all sources i = 1, 2, ...,N yields:

N∑i=1

λij =N∑i=1

(τijwi

ĀiLαi Pj

)1−σ= Pσ−1j

N∑i=1

(τijwi

ĀiLαi

)1−σ= Pσ−1j P

1−σj = 1

where the penultimate equality comes from the formula of the price index:

P1−σj =∑

j

( τijwiĀiLαi

)1−σI Multiply both sides of the equation by wjLj :

wjLj

N∑i=1

λij = wjLj ⇒N∑i=1

λijwjLj = wjLj



I Now, for both sides of the equation, take the summation across all j ’s:

N∑j=1

N∑i=1

λijwjLj =N∑j=1

wjLj

I On the left-hand side, invert the order of the summations:

N∑i=1

N∑j=1

λijwjLj =N∑j=1

wjLj (7)

I By hypothesis, we know the following equation holds for i = 1, 2, ...,N − 1:

wiLi =∑j

λijwjLj

I So it seems the term∑N

j=1 λijwjLj in equation (7) can be substituted by wiLiI But be careful! That only works for i from 1 to N − 1



I So, before we can use the hypothesis, let’s break ‘‘∑N

i=1’’ into two partsI One part is from i = 1 to i = N − 1. The other part is just the final term i = NI Equation (7) then becomes:( N−1∑

i=1

N∑j=1

λijwjLj

)+

N∑j=1

λNjwjLj =N∑j=1

wjLj

I Now we can apply the hypothesis to substitue wiLi for∑N

j=1 λijwjLj within thefirst summation: ( N−1∑

i=1

wiLi

)+

N∑j=1

λNjwjLj =N∑j=1

wjLj

I Rearranging terms:

N∑j=1

λNjwjLj =N∑j=1

wjLj −N−1∑i=1

wiLi



I Now let’s open up the terms in the right-hand side:

N∑j=1

λNjwjLj = (w1L1 + w2L2 + ...+ wN−1LN−1 + wNLN)

−(w1L1 + w2L2 + ...+ wN−1LN−1)I Cancelling terms on the right-hand side, we finally get:

N∑j=1

λNjwjLj = wNLN

I That is precisely the feasibility condition (equation (1)) applied to the NthlocationI That is, we have proved that the Nth market is also in equilibrium!I That concludes our proof


The Economics of Space 433: Section 5 Normalization and Log-Linearization · 2020. 2. 28. ·...

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